The singularities of the function \( \chi _{B,i}({\xi })\)  appear as concentrated poles of the Padé-approximant in the Borel plane and they depend on the dimensionless parameter \( \alpha \)  and \( \beta \) . In Fig. REF , we show the pole structure of both sectors in the Borel plane.
The poles start from a point nearest to the origin, namely \(\xi _{1,0}\)  and \(\xi _{2,0}\)  for \(\chi _{B}(\xi )\)  and \(\tilde{\chi }_{B}(\xi )\)  respectively, and accumulate along the real axis giving an image of branch cut as in the case of decoupled MIS [1]}. The starting value of the branch cut is proportional to the inverse of the slope of the coefficients shown in Fig. REF  and can be identified with the non-hydrodynamic mode that decays exponentially at late-times. From the analysis in Sec. REF  we readily see that \( \xi _{1,0} = \xi _{2,0} =\frac{ 3 }{2 C_{\tau }} \Big (\frac{\beta }{ \sqrt{\alpha \gamma }} \Big )^{1/3}= 3\epsilon _{10}^{1/4}\tau _0^{1/3} /(2 C_{\tau })=\xi _0 \)  when the condition (REF ) is satisfied, and \( \xi _{1,0} = \xi _{2,0} =\frac{ 3 }{2 C_{\tau }} \Big (\frac{\alpha }{ \sqrt{\beta \gamma }} \Big )^{1/3}= 3\epsilon _{20}^{1/4}\tau _0^{1/3} /(2 C_{\tau })=\xi _0 \)  when the condition (REF ) holds.
